// Numbas version: exam_results_page_options {"name": "Simon's copy of Find unit vector orthogonal to two others,", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variables": {"a": {"definition": "cross(u,v)", "name": "a", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "unitapos": {"definition": "apos/len(a)", "name": "unitapos", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "unita": {"definition": "a/len(a)", "name": "unita", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "u": {"definition": "vector(repeat(random(-9..9 except 0),3))", "name": "u", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "v": {"definition": "vector(repeat(random(-9..9 except 0),3))", "name": "v", "description": "", "templateType": "anything", "group": "Ungrouped variables"}, "apos": {"definition": "if(a[0]<0,-a,a)", "name": "apos", "description": "", "templateType": "anything", "group": "Ungrouped variables"}}, "advice": "

Note that in this advice, the full calculator display is used in the calculation of each step; any rounding is purely for display clarity.

A vector $\\boldsymbol{a}$, which is orthogonal to both $\\boldsymbol{u}$ and $\\boldsymbol{v}$, is given by the vector product:

\\[ \\boldsymbol{u}\\times\\boldsymbol{v}=\\pmatrix{u_2 v_3 - u_3 v_2, & u_3 v_1 - u_1 v_3, & u_1 v_2 - u_2 v_1} \\]

The magnitude of $\\boldsymbol{a}$ is given by

\\[ \\lvert\\boldsymbol{a}\\rvert=\\sqrt{a_1^2+a_2^2+a_3^2} \\]

A unit vector $\\boldsymbol{\\hat{a}}$ is obtained by dividing the components of the vector $\\boldsymbol{a}$ by its magnitude, i.e.

\\[ \\boldsymbol{\\hat{a}}=\\frac{\\boldsymbol{a}}{\\lvert\\boldsymbol{a}\\rvert} \\]

In this question,

\\[ \\boldsymbol{u \\times v} = \\pmatrix{\\simplify[basic]{{u[1]}*{v[2]}-{u[2]}*{v[1]}}, & \\simplify[basic]{{u[2]}*{v[0]}-{u[0]}*{v[2]}}, & \\simplify[basic]{{u[0]}*{v[1]} - {u[1]}*{v[0]}}} = \\var[rowvector]{a} \\]

and

\\[ \\lvert\\boldsymbol{u \\times v}\\rvert = \\sqrt{(\\var{a[0]})^2+(\\var{a[1]})^2+(\\var{a[2]})^2} = \\var{precround(len(a),3)} \\text{ to 3 decimal places.} \\]

A vector orthogonal to both $\\boldsymbol{u}$ and $\\boldsymbol{v}$ is therefore $\\boldsymbol{\\hat{a}}=\\frac{\\boldsymbol{u \\times v}}{\\lvert\\boldsymbol{u \\times v}\\rvert} = \\frac{1}{\\var{precround(len(a),3)}}\\var[rowvector]{a} = \\pmatrix{\\var{precround(unita[0],3)}, & \\var{precround(unita[1],3)}, & \\var{precround(unita[2],3)}}$ to 3d.p.

Remember that we were asked for the orthogonal vector with positive $x$-component. If our $x$-component of $\\hat{a}$ is negative, we can simply take the vector $-\\hat{a}$, which is the same unit vector but in the opposite direction.

Therefore our final answer is $\\pmatrix{\\var{precround(unitapos[0],3)}, & \\var{precround(unitapos[1],3)}, & \\var{precround(unitapos[2],3)}}$ to 3d.p.

", "metadata": {"description": "

Find a unit vector orthogonal to two others.

Uses $\\wedge$ for the cross product. The interim calculations should all be displayed to enough dps, not 3, to ensure accuracy to 3 dps. If the cross product has a negative x component then it is not explained that the negative of the cross product is taken for the unit vector.

", "licence": "Creative Commons Attribution 4.0 International"}, "functions": {}, "variablesTest": {"condition": "a[0]<>0", "maxRuns": 100}, "statement": "

Find the unit vector $\\boldsymbol{\\hat{a}}$, with positive $x$-component, which is orthogonal to both $\\boldsymbol{u}=\\pmatrix{\\var{u[0]},\\var{u[1]},\\var{u[2]}}$ and $\\boldsymbol{v}=\\pmatrix{\\var{v[0]},\\var{v[1]},\\var{v[2]}}$.

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$\\boldsymbol{\\hat{a}}=$ [[0]] (Enter your answers to 3d.p.)

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